Step 0: Skinning
You will need a sharp object. A knife, a screwdriver, even a shard of glass will do. Approach your target (which should be dead; if you're trying to skin a living being, something went horribly wrong) and use the tool on it. Skin as many times as you can, you usually get from 1 to 3 skin patches depending on mob size. Congrats, you successfully killed an innocent being for their leather. Fashion is blind to suffering, so lets continue.
Step 1: Dehairing
Using the same sharp object you need to dehair the skin. Simply use the tool on the skin to remove its hair. Repeat until you no longer have any hairy skin.
Step 2: Washing
Having successfully acquired dehaired skin, go to your nearest washing machine, they're generally on clothing storage, where the suit vendors are. Insert all of the dehaired skin into the Washing Machine, close the door and activate. The washing process will start, which will harden the leather.
Step 3: Drying
Once the washing process finishes, you are left with wet leather, which we'll need to dry. Leather dries if the tile the leather's on is at 40°C (313.15K/104°F) or higher. To dry the leather, you must drop the wet leather on the floor in a room and heat that room to at least 40°C, either via air alarm tampering, space heaters or fire. Wet Leather has a wetness value of 30 and loses 1 point per tick at drying temperature, so you'll need to sustain the heat for a while. Once the value reaches Zero you'll end up with Leather, which inherits the name of the skinned mob, ready to use.
The most common method is to just dump some welderfuel on the floor in a small room then set it alight with a lighter, matches or a welding tool since a small room with a welderfire can easily reach 1000+°C. A good room to do this on is on Art Supply Storage (Southwest of BoxStation's Tool Storage).
An alternative method is to acquire a space heater from maintenance, drop your wet leather on a tile with an airlock, drag the active heater to said airlock tile then close the airlock. Due to the heating being focused on only 1 tile it will rapidly go above 40°C.
Finally, if you have Air Alarm access, you can just raise the room's thermostat to 40°C. This is considerably slower than the other two methods, but it requires no further setup.
Step 4: Crafting
Ah, leather. A basic but very useful material. Just take it in your hand and use to craft.
Crafted Item Cost Time to Craft Bullwhip 10 10s Cowboy Hat 4 7s Cowboy Boots 4 8s Leather Gloves 2 9s Leather Shoes 4 8s Leather Satchel 12 13s Leather Wallet 4 9s Leather Helmet 3 9s Leather Armor 6 9s Leather Belt 3 6s 4x Leather Strip 1/20 (80x) 2s Ammunition Pouch 4 4s
Notes
The Leather belt can only hold 2 items but gives 4 extra punch damage which stacks with other punch modifiers (like Hulk or gloves) and raises your max HP by 4 when worn.
The bullwhip can be used as propulsion in space and is overall a very robust weapon against unarmored individuals.